CAPM - Assessment and Data Analysis

CAPM – Assessment and Data Analysis

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Table of Contents
Part A: Introduction ......................................................................................................................................................2
Early Components of CAPM......................................................................................................................................2
Evolution of CAPM ........................................................................................................................................................4
Empirical Performance of CAPM ........................................................................................................................5
Alternative Methods to CAPM.................................................................................................................................6
CAPM in the Modern Age ..........................................................................................................................................8
References ...........................................................................................................................................................................9
Part A: Appendix .........................................................................................................................................................10
Part B: Introduction....................................................................................................................................................11
Price Evolution of Apple and S&P500.............................................................................................................11
Statistical Analysis of Return...............................................................................................................................13
Explanation of Statistical Summary of Results ........................................................................................14
Risk-Return Analysis ...............................................................................................................................................17
Investigating the October Effect ........................................................................................................................19

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Part B: Appendix .................................................................................................................. 20
Part A - Introduction
Within this report I will be providing a framework in which I will be discussing the emergence
of the Capital Asset Pricing Model (CAPM) and how the model has evolved over time, while
discussing its empirical performance. To conclude, I will be giving my opinion on potential
alternative models to the CAPM and laying across the usefulness of the CAPM in modern
finance applications.
Early Components of Capital Asset Pricing Model
The CAPM was built upon the foundations of the Modern Portfolio Theory which was
developed by Henry Markowitz. Which is essentially how risk-adverse investors can construct
portfolios to maximize expected return based on a given level of risk (Markowitz, 1952), thus
highlighting that risk is a characteristic of higher reward. As Markowitz had noted, there is a
rate at which investors can gain expected return by taking on risk, or reduce risk by giving up
expected return (Markowitz, 1952). The CAPM is simply an example of an equilibrium model
in which asset prices are related to the exogenous data, the tastes and endowments of investors
(Brennan, 1989). The CAPM attempts to quantify the relationship between the beta of an asset
and its corresponding expected returns (Womack et al 2003). The difference between the
CAPM and alternative methods (which will be explained later) is such that the CAPM
incorporates the presence of a single risk (systematic risk) factor. While alternative methods
such as the Arbitrage Pricing Theory and the Fama French Model are examples of multi-factor
models. The CAPM is built around three main assumptions, which are as follows (for a full list
of CAPM assumptions see Part A- Appendix):
1. Investors’ concerns are only targeted towards expected return and the level of risk. Thus
assuming investors are completely rational when maximizing expected return for a
given level of risk (Womack et al 2003). In my opinion the idea of investors being
completely ‘’rational’’ when faced with the issue of wealth maximizing is not
necessarily true. As the field of behavioural finance provides an alternative way of
thinking, such that investors are not rational when faced with wealth maximizing.

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Theories such as Gambler’s Fallacy, Over-Confidence or Mental Accounting have
disputed CAPM’s claim of investors acting completely rational.
2. The second assumption is that all investors have the homogenous beliefs with concerns
about the risk and reward in the market.
3. The third assumption is that only one risk factor (systematic risk) is common to a
diversified market portfolio, which is in stark contrast to alternative methods.
Volatility and diversification have a strong role to play in the modern portfolio theory,
subsequently play an important role in defining the CAPM. Volatility is a measure of
dispersion of returns for a given asset or portfolio, with the general ruling being the higher the
volatility, the riskier the asset. Through the usage of diversification, a portfolio’s volatility to
some extent can be reduced. Since it has been established that investors are risk adverse and
prefer higher returns, such that through diversification (investors may opt for negative
covariance) they have adopted a strategy that allows them to decrease risk, while to some extent
not compromising expected return.
The CAPM starts out with two main ideas concerning the type of risk an investment is subject
to, namely, systematic risk and unsystematic risk. Unsystematic risk is reduced to some extent
by having a well-diversified portfolio, however systematic risk or market risk cannot be
diversified away due to the fact it’s a type of risk that affects the whole stock market or industry.
A practical example of market risk, is changes in interest rate. As I mentioned earlier, CAPM
is a one-factor model incorporating systematic risk, but it is indeed beta which represents this
risk within the CAPM. Beta is the measure of volatility of a security into comparison with the
market as a whole (BJS 1972). Within the CAPM a risk-free rate of return is also present, which
represents the level of interest an investor would expect from a risk free investment over a
specified period of time. The relationship between the Beta and Risk Free Rate of Return is
such that the CAPM predicts the expected return on an asset above the risk-free rate is
comparative to the non-diversifiable risk. In this sense the higher the quantity of beta of a
security, the higher is the expected return of that asset. It is key to remember that the CAPM
communicates to the investor by calculating the expected return by taking into account the risk
factor, such that if this expected return does not adequately compensate the investor for taking
greater risk with including the risk free rate, investors should not invest. Such that when
combining the following elements explained until now, we are left with the CAPM formula.

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The CAPM lives by the fact that an asset is expected to earn the risk-free rate plus a recompense
for bearing risk as measured by that asset’s beta.
Evolution of the Capital Asset Pricing Model
Since the induction of the CAPM it had been subject to numerous additions, namely by
financial economists such as Jack Treynor, William Sharpe, John Lintner and Fischer Black.
Sharpe (1964) and Linter (1965) had published papers regarding the prices of securities under
different conditions of risk. Within these papers, they simply wanted to explore the relationship
between higher risk and expected return, furthermore how to distinguish the part of the risk
that the market values. Sharpe had noted the element of diversification, such that some of risk
present in an asset can be avoided so that its total risk is obviously not the relevant influence
on its price (Sharpe 1964). Sharpe and Linter extended assumptions of Markowitz’ model with
the notion that all investors can borrow and lend an unlimited amount at an exogenously give
a risk-free rate of interest. Such that Sharpe and Lintner added two key assumptions to the
Modern Portfolio Theory in order to identify a portfolio that account for being a positive mean-
variance relationship (which is simply weighing risk against expected return). The first
assumption is that all investors are assumed to choose mean-variance efficient portfolios. The
second assumption is that there is borrowing and lending at a risk-free rate (Fama and French
2004), which does not depend on the amount borrowed or lent. With the first key assumption
complete agreement occurs, where investors agree on joint distribution of asset returns (Fama
and French, 2004). With this complete agreement investor would see the same opportunity set,
and they syndicate the same risk tangency portfolio with risk-free lending or borrowing.
Furthermore, by the equilibrium of asset market, since every investor holds the same risky asset
the tangency of the portfolio must be the value-weight market portfolio of risky assets. With
the second assumption of unlimited borrowing and lending at risk-free rate Sharpe-Lintner had
found where there is risk-free borrowing and lending, the expected return on assets that are
uncorrelated with the market return, must equal the risk-free rate (Sharpe, 1964). Sharpe and
Linter’s works laid the early foundations of the CAPM. By combining the earlier components
of the Markowitz’s Theory in this paper and additions made by Sharpe and Lintner, we are left
with the CAPM equation:
𝐸𝑅𝑖 = 𝑅𝑓 + (𝐸𝑅𝑚 − 𝑅𝑓) 𝛽𝑖

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Concluding the two key assumptions added by Sharpe and Lintner, was such that there was
confusion surrounding risk-free borrowing and lending as an unrealistic assumption. This is
where financial economist Fischer Black (1972) developed another variety of the CAPM,
known as the Black CAPM. Black (1972) developed a version of the CAPM without risk-free
borrowing or lending (Fama and French, 2004). Although, Black’s key results were such that
the market portfolio is mean-variance efficient (much like Sharpe-Lintner), instead Black
allowed for unrestricted short sales of risky assets.
Empirical performance of CAPM
During the past decades, there has been numerous empirical evidence that has
supported/targeted some of the basic assumptions of the CAPM, with many quarters calling it
to some extent practical but also unrealistic. One important study conducted on the CAPM was
that of Black, Jensen and Scholes (BJS, 1972). BJS study was conducted on sample size of all
securities listed on the NYSE for the period 1926-1966, using percentage monthly returns. BJS
came up with a strategy in which they created portfolios with very different betas for use in
empirical test (Jagannathan et al 1995). Methodology which was employed by BJS was that of
a ‘’Two Pass Methodology’’. The first pass involved estimating historical beta using a time
series regression (Jagannathan et al 1995). The equation below shows the relationship between
estimating the beta using a time regression in conjunction with excess returns on the market
returns.
For the first pass test BJS estimated the beta for individual security using monthly returns for
the 5-year period 1926-1930 (BJS, 1972). In which ten portfolios were ranked according to the
estimated beta (high to low). They then calculated the monthly return for each portfolio for
1931. BJS repeated these phases numerous times, with the results of these process exhibiting a
series of monthly returns for 10 portfolios. Moreover, for the 35-year period BJS calculated
the mean monthly return and estimated the beta coefficient for each of the 10 portfolios. Lastly,
BJS regressed the mean portfolio returns against the portfolios. The result of the BJS test was
such that they find the data is consistent with the predictions of the CAPM, given the fact that
the CAPM is an estimate to actuality just like any other model (Jagannathan et al 1995). The

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results indicated that the relationship between average return and beta is indeed close to a linear
relationship, and that portfolios with a high (low) beta have high (low) average returns, this
positive relationship is one in which the CAPM depicts. Ultimately, the BJS study was in
favour of the CAPM.
Fama and MacBeth (1973) study examined whether there is a positive linear relationship
between average return and beta and whether the squared value of beta and the volatility of the
return on an asset can explain the persistent variation in average returns across assets that is
not explained by risk factor alone. (Jagannathan et al 1995). Sample size for FM study was all
common stock traded on the NYSE for the period Jan 1926-June 1968. Like the BJS study, the
FM too was a ‘’Two Pass Methodology’’. Much like the BJS study, the results from FM (1973)
lent support to the CAPM. Mainly Fama and MacBeth (1973) highlighted the evidence of a
larger intercept term than the risk-free rate, the linear relationship between the average return
and the beta holds and lastly that the linear relationship holds well when the data covers a long
time. However, in my opinion, subsequent studies have provided evidence of a weak
relationship in the CAPM. For instance, Fama and French (1992) had come to the conclusion
that stock betas (risk) alone did not explain long term relationships, although a combination of
SMB and HML factors did. Furthermore, Ross (1976) published a paper on the Arbitrage
Pricing Theory (APT). With the APT it does not require restrictive assumptions as does the
CAPM. The CAPM model since its inception, as gone under the scanner with reasonable
success being found in its support through studies carried out by BJS (1972) and Fama and
MacBeth (1973). On the other hand, in my opinion academics realized the limitations of the
CAPM being a one-factor model, with numerous factors better able to explain the relationship
between risk and expected return. Thus, this saw the emergence of competing models such as
the Fama-French Three Factor Model and Arbitrage Pricing Theory, which have to some extent
redefined the very basic assumptions of the CAPM.
Alternative Methods to CAPM
Within this section I will be briefly explaining the fundamentals of the most widely thought
alternatives to the CAPM, known as the Fama and French Three Factor Model and the
Arbitrage Pricing Theory.

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The Fama and French model was developed by financial economist Eugene Fama and Ken
French. The Fama and French Three Factor Model was developed in response to heavy
criticism the CAPM was receiving due to performing poorly in realised returns. In stark
contrast to the CAPM which was a one-factor model (incorporating risk), the Fama and French
added two additional factors, namely value and size. To represent these two additional risks,
Fama and French had constructed SMB (Small Minus Big) to represent size and HML (High
Minus Low) to represent value (Womack et al 2003). Fama and French had come to this
conclusion of incorporating size and value due to research they had conducted, with findings
telling the story of small cap companies and value stocks outperform large cap and growth
stocks. SMB is designed to measure the additional returns investors have historically received
by investing in stocks with small cap. Such that by subtracting the average return of the smallest
30% of stocks from the average return of the largest 30% of stocks, one would get the SMB
value. With a positive SMB highlighting small cap companies outperform large cap companies.
While the HML factor suggests higher risk exposure for typical ‘’value’’ stocks (High B/M)
versus ‘’growth’’ stocks (Low B/M) (Womack et al 2003). The HML factor is calculated much
the same as SMB, the average return of 50% of stocks with the highest B/M minus the average
return of the 50% of stocks with the lowest B/M. With a positive HML indicating that value
stocks outperformed growth stocks in that given month, HML was computed. Moreover, by
combining the systematic risk factor of the CAPM, with value and size risk we are left with
the following equation:
Beta is still considered as a measurement to the market risk, while sA measures the level of
exposure to size risk, with hA measures the level of exposure to value risk. Within financial
academic works, the Fama and French Three Factor Model has seen considerable success, as
they are believed to have the greatest predictive power of any of two additional factors that
research has tested (Womack et al 2003). Furthermore, running a regression using the three
factor model often yields a R2 value of 0.95 if compared with CAPM only yielding usually R2
value of 0.85. Principal uses of the Fama and French Model is to classify mutual funds and
separating the different funds to allow investors to weight their risk exposure. When comparing
the CAPM and Fama French Model it is important to put both models to an empirical test. Such
that a test was carried out in emerging markets by (Al-Mwalla et al 2012) on the Amman Stock
Market, during the period June 1999 – June 2010. When using the CAPM the study did not

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find any evidence that support the ability of the single factor model to provide a stable
explanation to the variation in portfolios rates of returns (Al-Mwalla et al 2012). On the other
hand, using the FF Model it explains a good part of variation in stock return, but not all of it
which means that there are other variables to explain stock returns. However, the FF model has
more explanatory power than the CAPM.
The Arbitrage Pricing Theory (APT) was a model developed by Stephen Ross, which looks to
explain the approach to portfolio strategy decision which involves choosing the desirable
degree exposure to the central economic risks that effect both asset returns and organizations
(Roll and Ross, 1995). APT promotes the idea of returns being broken down into expected and
unexpected returns. The APT is seen to be a more relaxed model to the CAPM such that it
allows for multiple factors but in a more practical way in the sense that it does not assume all
investors implement Markowitz portfolio selection methods. The systematic factors his
research has identified as the most important are: (1) unanticipated inflation (2) changes in
expected level of industrial production (3) shifts in risk premiums (4) movements in interest
rates. (Ross and Roll, 1995), thus it makes the use of macro-economic variables. Using the law
of one price portfolio investors should be able to construct portfolios which are risk free, as
sensitivities in different stock cancel each other out.
CAMP in the Modern Age
Since the introduction of the CAPM decades ago, it has indeed gone under the scanner, with
additions being made, criticism, and alternative methods being proposed. Such that it begs the
question is the CAPM still useful in the modern age?
In my opinion the CAPM is still important to some extent, such that it is still widely taught and
used in the investment community. Using the CAPM an investor can still be assured that a
stock with a high/low beta will tell the story of how risk/less risk a particular asset is when
comparing it with the market. One obvious advantages of the CAPM is such that it is generally
seen as a much better model of calculating cost of equity than the dividend growth model, such
that it takes into account a stock’s level of market risk relative to the stock’s market as a whole.
More so the CAPM can still be used in the modern age for appliances such as portfolio
management, evaluating portfolio managers and cost of capital determination. In my opinion,
CAPM still have a place in modern finance applications because of the model’s ability to

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quantify risk, as well as explaining risk measures into estimates of expected returns. However,
it should not be solely relied upon, because of the shortcomings as highlighted previously. As
the famous Robert D. Arnott once said ‘’in investing what is comfortable is rarely profitable’’.
References
 Al-Mwalla, M., Al-Qudah, K. and Karasneh, M. (2012). Addtional Risk Factors that
can be used to Explainmore Anomalies: Evidence from Emerging Market. International
Research Journal of Finance and Economics, -(99), pp64-74.
 Black, F. (1972). Capital Market Equilibrium with Restricted Borrowing. The Journal
of Business, 45(3), p.444.
 Black, F., Jensen, M. and Scholes, M. (1972). The Capital Asset Pricing Model: Some
Empirical Tests. Studies In The Theory of Capital Markets, Praeger Publishers, -(-),
pp.1-54.
 Brennan, M.J., "capital asset pricing model", "The New Palgrave Dictionary of
Economics", Eds. Steven N. Durlauf and Lawrence E. Blume, Palgrave Macmillan,
2008, The New Palgrave Dictionary of Economics Online, Palgrave Macmillan. 06
April 2016, DOI:10.1057/9780230226203.0190
 Fama, E. and French, K. (1992). The Cross-Section of Expected Returns. The Journal
of Finance, 47(2), pp.427-458.
 Fama, E. and French, K. (2004). The Capital Asset Pricing Model: Theory and
Evidence. The Journal of Economic Perspectives, 18(3), pp25-46.
 Fama, E. and MacBeth, J. (1973). Risk, Return, and Equilibrium: Empirical Tests.
Journal of Political Economy, 81(3), pp.607-636.
 Jagannathan, R. and McGrattan, E. (1995). The CAPM Debate. Federal Reserve Bank
of Minneapolis Quarterly Review, 19(4), pp.pp2-17.
 Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), pp.77-89.
 Roll, R. and Ross, S. (1995). The Arbitrage Pricing Theory Approach to Strategic
Portfolio Planning. Financial Analysts Journal, 51(1), pp.1-7.
 Ross, S. (1976). The arbitrage theory of capital asset pricing. Journal of Economic
Theory, 13(3), pp.341-360.
 Sharpe, W. (1964). Capital Asset Prices: A Theory of Market Equilibrium under

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Conditions of Risk. The Journal of Finance, 19(3), p.425-442
 Womack, K. and Zhang, Y. (2003). Understanding Risk and Return, the CAPM, and the
Fama-French Three-Factor Model. Tuck School of Business, 3(111), pp.1-14.
Part A -Appendix
Full list of CAPM assumptions
1. Investors are risk adverse
2. Investors seek maximizing expected return
3. Investors have homogeneous expectations
4. Borrow or Lend freely a risk less rate of interest
5. Market is perfect
6. Quantity of risk securities in market is given
7. No transaction costs

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Part B - Introduction
Apple Inc. (AAPL) was founded by Steve Jobs, Steve Wozniak and Ronald Wayne on April
1st 1979, with the company being involved within the computer industry. Apple underwent its
IPO on December 12, 1980 at $22.00 per share. Since the Apple listing the stock as split 4
times, with the first split taking place June 16, 1987 (2 for 1 split), June 21, 2000 (2 for 1
split), February 28, 2005 (2 for 1 split) and June 09, 2014 (7 for 1 split). Apple stock has
undergone ups and downs in terms of stock growth. Apple’s stock performance has a close
link with the relevance the general public sees with its products. Such that during the era of
the Macintosh, stock performance languished due to the general public not finding any
relevance with the product, which had a direct impact on share performance. When original
co-founder Steve Jobs returned to Apple, after originally being ousted implemented a new era
of innovation within their production lines. In the late 1990’s stock performance was very
respectable. Emergence of the iPhone and other related products completely transformed
Apple has a viable and profitable company, such that share price rallied and continued to
grow, even after Steve Jobs’ death. Currently, Apple’s share price can be described as steady
even though iPhone sales are sluggish. All data used in this analysis are lognormal returns.
Price Evolution of Apple and S&P500
Figure 1

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Above Figure 1 and Figure 2 represent the price evolution of both Apple’s stock price and the
S&P500 index. Figure 1 shows the trading volume of Apple (in millions) on its primary
vertical axis, while showing Apple’s stock price as well. Looking at the relationship between
0
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Millions Apple Stock Price and Volume Chart 2000-2015
Volume Close
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APPLE AND S&P500 PRICE EVOLUTION
S&P500 Apple
Figure 2

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the trading volume and stock performance of Apple it seems to be from 2000 onwards trading
volume has exceeded Apple’s stock performance. Specifically, during the period form 2005-
late 2008, up until the financial crisis had hit. Although during the financial crisis, overall
trading volume declined, Apple’s share price performance improved, which concedes with
the emergence of innovative products such as the iPhone. Figure 2 represents the price
evolution of Apple and S&P500. Following the financial crisis in 2008, both Apple and the
S&P 500 took a sharp dip. In 2008, Apple shares fell more than 50%, however since as it can
be seen from Figure 2, Apple has been consistently risen 5% or more. Although Apple has
been consistently improving since the financial crisis, it was reported in last quarter of 2015
that Apple was on course for having its worst year since the financial crisis, even though it
has done better than the broader market. Like Apple, the S&P 500 took a significant hit from
the onset of the financial crisis. The index fell 56.8% from its peak on October 2007 to a low
point on March 2009. Unlike Apple, which had a consistent increase in stock performance,
the S&P 500 increased rapidly, ending 2013 with significant gains. Most recently, as oil
prices continue to slump among other aspects has had a significant impact on the S&P 500
performance for the close of 2015, with S&P 500 like Apple enduring its worst year since
2008. The S&P 500 ended the year down 0.73% after three-straight years of double-digit
gains.
Statistical Analysis of Returns
Figure 3 Figure 4

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Within this section I will be focusing on the statistical analysis of returns when using the
frequency of weekly returns, but also comparing how the statistical analysis of return changes
when using different frequencies. Referring to the Summary Results of Statistics (Figure 5)
when applying weekly returns Apple possesses a skewness value of -2.31 and S&P500 -0.82.
Both and Apple and S&P500 are negatively skewed (skewed to the left), with an
asymmetrical distribution (See Figure 3+4 above). Referring back to Figure 5 it is important
to point out that as the frequency of returns are increasing skewness for both Apple and
S&P500 are decreasing, becoming more symmetrical. Nevertheless, whatever frequency is
being applied the skewness value is below 0. Illustrating distributions that investors call a
long left tail (See Figure 5 Weekly Returns). Which allows for an investor for a greater
chance of extremely negative/positive outcomes. Using the descriptive statistics tool within
excel, I was able to compute a variety of summary results for both Apple and S&P500 across
all frequencies (See Part B - Appendix). Kurtosis for weekly frequencies was calculated as
being 25.39 for Apple and 6.91 for S&P500. However, not just for weekly returns, but across
all frequencies kurtosis for Apple is higher than the market index. In Apple’s case the
kurtosis is higher than three across all frequencies, which means they are said to be
leptokurtic. With kurtosis being leptokurtic, this fat tail means the distribution is more
clustered around the mean than in mesocratic distribution. Lastly adjusting the frequency of
returns brings along with it a reduction in kurtosis for both Apple and the market index.
Explanation of Statistical Summary of Results
Figure 5

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Within this section I will be examining the returns estimated by CAPM and carrying out
linear regressions. I will also be showcasing how the changing of frequencies affect the beta
overtime.
Referring to the table above, the beta has stayed fairly static with a small decrease when
changing frequencies from daily to weekly. However, the beta increased significantly when
applying monthly returns. Using the regression analysis tool in excel I was able to calculate
the beta of Apple in which I was able to assess how risky Apple is in line with the S&P 500.
With beta being a suitable measure of the volatility of an individual security or a portfolio in
comparison to the market as a whole. In the case of weekly returns of Apple, the beta amounted
to 1.10, which tells me that Apple is slightly more volatile than the S&P 500 (market beta of
1), furthermore the beta of 1.10 tells me Apple is 10% more volatile than the S&P 500. Looking
at the figures as an investor, I would quickly realize that a beta of over 1 is considered the norm
among high tech companies listed on the S&P 500, secondly with a higher beta provides me
as the investor an opportunity for greater return by accepting this higher element of risk. Using
descriptive statistics which provided me with a summary of results, concerning risk and return.
Using the data results, I am able to use standard deviation as another illustration of volatility.
In the case of Apple had a standard deviation of approximately 5.9% which is seen to be more
volatile than the overall market, S&P 500, with a standard deviation of 2.53%. Referring to
Figure 1, Apple had high volatility in terms of volume when the dot-com bubble reached its
climax, although it seems Apple’s price was not affected greatly by this speculation, on the
other hand S&P 500 declined has dot-com bubble reached its climax.
When taking beta into account as a measure volatility which is a central competent of the
CAPM, I investigated the results of the regression analysis. Firstly, regarding to the summary
results of the regression analysis of weekly returns (See Part B- Appendix), I found the
following. Interpreting the R2 Value of Apple, which amounted to 22%. In my understanding,
the R2 value explains the percentage of the risk-return relationship within CAPM. Such that
with 22% of the stock’ performance is explained by its risk exposure, as measured by beta.
The higher the R2 the better, as it simply tells the story that the CAPM explains majority of
risk exposure, however not in the case of Apple and S&P 500. Additionally, corresponding to
Linear Graph (see below) the trendline tells the story of how accurate the CAPM is, with a
perfect trendline being one that risk exposure points lie on the trendline, which would exhibit
a R2 of 1.0 or 100%. As you can see from the CAPM Graph many points are dispersed which
tells me in this case, the CAPM does not tell a significant amount of the stock’s performance.

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To examine the mean, I will be applying the mean reversion theory to Apple, which simply
says that prices and returns eventually move back towards the mean. Such that investors can
utilize the mean to predict future return, in the sense that as prices deviate from the mean they
will eventually fall back to the mean, giving the investor a chance to either profit by buying
or profit/limit loss by selling before prices/returns retreat to the mean. Apple’s monthly return
mean during the period 2000-2015 amounted to approximately 1.79%. In the case of Apple,
monthly returns for 2009 was 8%, which saw a decrease to 4% by 2010, with returns
eventually retreating towards the mean of 2% in 2011 and 2012, before picking up again in
2014% with a mean of 4%. In terms of volatility, referring to the statistical table above
y = 1.1007x + 0.0037
R² = 0.2209
-0.8
-0.6
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0
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-25.00% -20.00% -15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%
Weekly Apple Returns
rApple
Linear (rApple)
-100.00%
-80.00%
-60.00%
-40.00%
-20.00%
0.00%
20.00%
40.00%
60.00% Mean Reverting Process 2000-2015
rS&P500 rApple Linear (rS&P500) Linear (rApple )
Figure 6

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(Figure 5), it is clear Apple is indeed more volatile than the market overall, with an increase
volatility as frequency changes from daily to monthly. Apple experienced high amount of
volatility during the recession years (2007-2010), with high/low swings of stock movement.
See Appendix for graphical representation.
Risk-Return Analysis
7 Year Analysis
Expected
Return
Actual
Return R2 Value
Percentage
Difference Conclusion
2015 -0.06% -4.53% 50.24% 98.67% Overpriced
2014 10.17% 38.27% 23.03% 73.42% Underpriced
2013 9.99% 6.38% 1.88% 36.00% Overpriced
2012 16.64% 23.01% 27.52% 27.68% Underpriced
2011 -1.00% 18.64% 39.94% 94.63% Underpriced
2010 12.74% 41.98% 60.98% 69.65% Underpriced
2009 22.17% 84.43% 53.69% 73.74% Underpriced
2008 -32.171% -64.3% 27.48% 48.44% Overpriced
In the first part of this section I will be covering the period 1st January, 2008 to 31st December
2015, of weekly Apple returns comparing CAPM results with actual stock return results. The
methodology which I undertook was based upon first firstly, the estimation of the systematic
risk beta of Apple relation to S&P 500; secondly, the estimation of market risk premium of
the model with regards to the market; and lastly, to test whether the model can explain the
relationship between individual stock return and systematic risk, beta. Based on my findings
majority of expected return produced by the CAPM seem to be under-priced when compared
Figure 7

18 | P a g e
with actual returns of Apple. For example, during 2012 the actual stock return is 23.01%
while the model predicted a return of 16.64% with a beta of 1.2. Thus the actual return was
27.68% higher than predicted. Similarly, the actual returns exceeded expected returns (as
predicted by the CAPM) by 69.65%, 73.74% and 73.42% in 2010, 2009 and 2014
respectively. Such that if investors had contemplated to invest in these given years they
would have got a bargain since the stocks were undervalued in those years. Moreover,
investors would have been more than compensated for taking additional risk. Thus
highlighting investors are well compensated for this relatively high beta figures. Yet, 2013
the CAPM results indicated that the stock was overpriced. With the CAPM predicting an
expected return of 9.99% with an actual stock return of 6.38%. Moreover, in 2008 at the
height of the financial crisis, most stocks and markets overall were going downwards. In
2008, with a beta of 0.78 calculated an expected return of -32.17%, however it was grossly
inaccurate as Apple’s stock declined by -64.3%. From my findings, I conclude that the
CAPM cannot be used to statistically explain the observed differences in the actual and
expected return on the Apple stock. The implication is that, the observed differences in the
variables in the actual and the predicted returns are statistically insignificant and likely due to
chance or other factors and not due to the systematic risk factors as measured by beta of the
Apple stock under review.
7 Year Analysis CAPM Return
Beta
CAPM
Expected
Return
Apple
Return Market Return
2015 1.3657 -0.06% -4.53% -0.04%
2014 0.9181 10.17% 38.27% 11.08%
2013 0.4574 9.99% 6.38% 21.84%
2012 1.2086 16.64% 23.01% 13.77%
2011 0.911138 -1.00% 18.64% -1.10%
2010 1.357 12.74% 41.98% 9.38%
2009 0.984822 22% 84% 23%
2008 0.788817 -32% -64% -41%
Figure 8

19 | P a g e
Within the second part of this section I will be analysing how the beta has changed during the
last 7 years for Apple and what effect a change in beta had on the required rate of return as
predicted by the CAPM, to investigate if the formula holds up, while comparing CAPM
results with market return. The cornerstone of the CAPM simply says that the only reasons
investors should earn more, is by taking additional risk. With the end result of the CAPM
formula giving investor a picture of the expected return they should expect for a given level
of risk. Such that if the expected return is not sufficient, investors should not invest. During
2012, Apple had a beta of 1.20 with a required rate of return of 16.64%. With this
considerable amount of risk an investor would expect Apple to outperform the market (which
has a market beta of 1). Indeed, Apple outperformed the market by achieving returns of
16.64% compared to 13.77% of that of the S&P500. However, in 2011 Apple had a beta of
0.91 which is still relatively high, in line with the market beta. With this beta in mind an
investor would expect Apple return to be in line with the market return. Nevertheless, Apple
achieved a return of 18.64% with the S&P500 achieving -1.10%, with CAPM estimates being
wide of the mark with a -1.00% expected return. This shows the inability of the CAPM to
account for other factors and shows the limitations of a one-factor model.
Investigating the ‘’October Effect’’
The October Effect is preconceived notion that stocks tend to decline during the month of
October. The October Effect is considered mainly to be a psychological anticipation rather
than an actual singularity. Such that I put this theory to the test when it comes to Apple.
Below Figure 9 showcases Apple’s September closing prices (October Opening) and October
closing prices (November Opening) during the period 2000-2015. As it can be seen from the
graph below October Prices tend to outperform during the month, thus dispelling the theory
of the October effect in Apple’s case.

20 | P a g e
Part B - Appendix
Monthly Returns Distribtuion
0
100
200
300
400
500
600
700
800
1998 2000 2002 2004 2006 2008 2010 2012 2014 2016
October Effect of Apple Disapproved 2000-2015
Sept Closing Oct Closing
Figure 9

21 | P a g e
Daily Returns Distribution
SUMMARY OUTPUT - Weekly Returns
Regression Statistics
Multiple R 0.470018368
R Square 0.220917267
Adjusted R Square 0.219979742
Standard Error 0.052425996
Observations 833
ANOVA
df SS MS F Significance F
Regression 1 0.647650164 0.647650164 235.6389645 5.2608E-47
Residual 831 2.28399105 0.002748485
Total 832 2.931641214
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 0.003692052 0.001816702 2.032283005 0.042442576 0.000126188 0.007257916 0.000126188 0.007257916
Beta 1.10067664 0.071702813 15.35053629 5.2608E-47 0.959936724 1.241416557 0.959936724 1.241416557

22 | P a g e
SUMMARY OUTPUT - Monthly Returns
R Square 0.240955025
Observations 191
ANOVA
Regression 1 0.809380357 0.809380357 59.99710335 5.6306E-13
Residual 189 2.549671216 0.013490324
Total 190 3.359051573
Intercept 0.014933303 0.00841286 1.775056628 0.07749804 -0.001661863 0.031528469 -0.001661863 0.031528469
Beta 1.479622262 0.191023024 7.745779712 5.6306E-13 1.102811186 1.856433339 1.102811186 1.856433339
SUMMARY OUTPUT - Daily Returns
R Square 0.250870464
Observations 4024
ANOVA
Regression 1 0.820786445 0.820786445 1346.897913 1.3602E-254
Residual 4022 2.450967553 0.00060939
Total 4023 3.271753998
Intercept 0.000735334 0.00038916 1.889539908 0.058891379 -2.76359E-05 0.001498303 -2.76359E-05 0.001498303
Beta 1.127649257 0.030726048 36.7001078 1.3602E-254 1.067409182 1.187889332 1.067409182 1.187889332

23 | P a g e
y = 1.4796x + 0.0149
R² = 0.241
-100.00%
-80.00%
-60.00%
-40.00%
-20.00%
0.00%
20.00%
40.00%
60.00%
-25.00% -20.00% -15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%
Monthly Apple Returns
rApple
Linear (rApple )
y = 1.1276x + 0.0007
R² = 0.2509
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%
Daily Apple Returns
rApple
Linear (rApple)

24 | P a g e
Weekly Returns Stats
rApple rS&P500
Mean 0.004153485 Mean 0.000419227
Standard Error 0.002056701 Standard Error 0.000878267
Median 0.007510853 Median 0.001599362
Standard Deviation 0.05935998 Standard Deviation 0.025348299
Sample Variance 0.003523607 Sample Variance 0.000642536
Kurtosis 25.39394458 Kurtosis 6.917077364
Skewness -2.30843959 Skewness -0.81627775
Range 0.942615312 Range 0.314396467
Minimum -0.7064082 Minimum -0.20083751
Maximum 0.236207111 Maximum 0.11355896
Sum 3.459853233 Sum 0.349215883
Count 833 Count 833
Monthly Returns Stats
rApple rS&P500
Mean 0.01789543 Mean 0.002001948
Median 0.02907319 Median 0.008322837
Range 1.235581828 Range 0.287943065
Minimum -0.861414003 Minimum -0.185636474
Maximum 0.374167825 Maximum 0.102306592
Sum 3.418027053 Sum 0.382372073
Count 191 Count 191
Daily Returns Stats
rApple rS&P500
Mean 0.000830534 Mean 8.4424E-05
Median 0.000769827 Median 0.000535677
Range 0.86144108 Range 0.204267093
Minimum -0.73124689 Minimum -0.094695125
Maximum 0.13019419 Maximum 0.109571968
Sum 3.342070376 Sum 0.339722217
Count 4024 Count 4024

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